Geometrical Phase, Generalized Quasienergy and Floquet Operator as Invariants

Abstract

Quantum mechanics is usually connected with energy spectra of the systems with stationary Hamiltonians.The dynamics of these systems is described by the transitions between the energy levels. The nonstationary quantum systems have no energy levels due to the absence of symmetry related to time displacements. But for periodical quantum systems there exists the symmetry corresponding to crystal time structure of Hamiltonian.Due to this the notion of quasienergy levels has been introduced in Ref.[1] and [2]. The main point of the quasienergy concept is to relate the quasienrgies to the eigenvalues of the Floquet operator which is equal to the evolution operator of a quantum system taken at a given time moment. The aim of this article is to relate the Floquet operator to integrals of motion and to introduce a new operator which is the integral of motion and has the same quasienergy spectrum that the Floquet operator has. Implicitly this result was contained in Ref.[3] but we want to have the explicit formulae for the new integral of motion.